<h2>DESCRIPTION</h2>

This numerical module calculates implicit transient and steady state,
confined groundwater flow in three dimensions based on volume maps
and the current 3D region settings.
All initial- and boundary-conditions must be provided as volume maps.
The unit in the location must be meters.
<p>This module is sensitive to mask settings. All cells which are outside
the mask are ignored and handled as no flow boundaries.
<p>The module calculates the piezometric head and optionally the water
balance for each cell and the groundwater velocity field in 3 dimensions.
The vector components can be visualized with ParaView if they are exported
with <em>r3.out.vtk</em>.

<p>The groundwater flow will always be calculated transient. 
For steady state computation the user should set the timestep
to a large number (billions of seconds) or set the
specific yield raster map to zero.

<h2>NOTES</h2>

The groundwater flow calculation is based on Darcy's law and a numerical
implicit finite volume discretization. The discretization results in a
symmetric and positive definite linear equation system in form of
<i>Ax = b</i>, which must be solved. The groundwater flow partial
differential equation is of the following form:

<p>(dh/dt)*S = div (K grad h) + q
<p>In detail for 3 dimensions:
<p>(dh/dt)*S = Kxx * (d^2h/dx^2) + Kyy * (d^2h/dy^2) + Kzz * (d^2h/dz^2) + q

<ul>
<li>h -- the piezometric head im meters [m]</li>
<li>dt -- the time step for transient calculation in seconds [s]</li>
<li>S -- the specific yield  [1/m]</li>
<li>b -- the bottom surface of the aquifer meters [m]</li>
<li>Kxx -- the hydraulic conductivity tensor part in x direction in meter per second [m/s]</li>
<li>Kyy -- the hydraulic conductivity tensor part in y direction in meter per seconds [m/s]</li>
<li>Kzz -- the hydraulic conductivity tensor part in z direction in meter per seconds [m/s]</li>
<li>q - inner source/sinc in [1/s]</li>
</ul>

<p>Two different boundary conditions are implemented, the Dirichlet and
Neumann conditions. By default the calculation area is surrounded by
homogeneous Neumann boundary conditions. The calculation and boundary
status of single cells can be set with the status map, the following
cell states are supported:

<ul>
<li>0 == inactive - the cell with status 0 will not be calculated,
 active cells will have a no flow boundary to an inactive cell</li>
<li>1 == active - this cell is used for groundwater calculation,
 inner sources can be defined for those cells</li>
<li>2 == Dirichlet - cells of this type will have a fixed piezometric
 head value which do not change over time </li>
</ul>

<p>Note that all required raster maps are read into main memory. Additionally
the linear equation system will be allocated, so the memory consumption of
this module rapidely grow with the size of the input maps.

<p>The resulting linear equation system <i>Ax = b</i> can be solved with
several solvers. An iterative solvers with sparse and quadratic matrices
support is implemented.
The conjugate gradients method with (pcg) and without (cg) precondition.
Additionally a direct Cholesky solver is available. This direct solver
only work with normal quadratic matrices, so be careful using them with
large maps (maps of size 10.000 cells will need more than one Gigabyte
of RAM). The user should always prefer to use a sparse matrix solver.

<h2>EXAMPLE 1</h2>
This small script creates a working groundwater flow area and
data. It cannot be run in a lat/lon location.

<div class="code"><pre>
# set the region accordingly
g.region res=25 res3=25 t=100 b=0 n=1000 s=0 w=0 e=1000 -p3

#now create the input raster maps for a confined aquifer
r3.mapcalc expression="phead = if(row() == 1 && depth() == 4, 50, 40)"
r3.mapcalc expression="status = if(row() == 1 && depth() == 4, 2, 1)"
r3.mapcalc expression="well = if(row() == 20 && col() == 20 && depth() == 2, -0.25, 0)"
r3.mapcalc expression="hydcond = 0.00025"
r3.mapcalc expression="syield = 0.0001"
r.mapcalc  expression="recharge = 0.0"

r3.gwflow solver=cg phead=phead statuyield=status hc_x=hydcond hc_y=hydcond  \
   hc_z=hydcond sink=well yield=syield r=recharge output=gwresult dt=8640000 vx=vx vy=vy vz=vz budget=budget

# The data can be visualized with ParaView when exported with r3.out.vtk
r3.out.vtk -p in=gwresult,status,budget vector=vx,vy,vz out=/tmp/gwdata3d.vtk

#now load the data into ParaView
paraview --data=/tmp/gwdata3d.vtk
</pre></div>

<h2>EXAMPLE 2</h2>
This will create a nice 3D model with geological layer with different
hydraulic conductivities. Make sure you are not in a lat/lon projection.

<div class="code"><pre>
# set the region accordingly
g.region res=15 res3=15 t=500 b=0 n=1000 s=0 w=0 e=1000

#now create the input raster maps for a confined aquifer
r3.mapcalc expression="phead = if(col() == 1 && depth() == 33, 50, 40)"
r3.mapcalc expression="status = if(col() == 1 && depth() == 33, 2, 1)"
r3.mapcalc expression="well = if(row() == 20 && col() == 20 && depth() == 3, -0.25, 0)"
r3.mapcalc expression="well = if(row() == 50 && col() == 50 && depth() == 3, -0.25, well)"
r3.mapcalc expression="hydcond = 0.0025"
r3.mapcalc expression="hydcond = if(depth() &lt; 30 && depth() > 23 && col() &lt; 60, 0.000025, hydcond)"
r3.mapcalc expression="hydcond = if(depth() &lt; 20 && depth() > 13 && col() >  7, 0.000025, hydcond)"
r3.mapcalc expression="hydcond = if(depth() &lt; 10 && depth() >  7 && col() &lt; 60, 0.000025, hydcond)"
r3.mapcalc expression="syield = 0.0001"

r3.gwflow solver=cg phead=phead statuyield=status hc_x=hydcond hc_y=hydcond  \
   hc_z=hydcond sink=well yield=syield output=gwresult dt=8640000 vx=vx vy=vy vz=vz budget=budget

# The data can be visualized with paraview when exported with r3.out.vtk
r3.out.vtk -p in=gwresult,status,budget,hydcond,well vector=vx,vy,vz out=/tmp/gwdata3d.vtk

#now load the data into paraview
paraview --data=/tmp/gwdata3d.vtk
</pre></div>

<h2>SEE ALSO</h2>

<em>
<a href="r.gwflow.html">r.gwflow</a>,
<a href="r.solute.transport.html">r.solute.transport</a>,
<a href="r3.out.vtk.html">r3.out.vtk</a>
</em>

<h2>AUTHOR</h2>
S&ouml;ren Gebbert
<p>This work is based on the Diploma Thesis of S&ouml;ren Gebbert available
<a href="https://grass.osgeo.org/gdp/hydrology/gebbert2007_diplom_stroemung_grass_gis.pdf">here</a>
at Technical University Berlin, Germany.

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